Stability of self-similar solutions in a simplified LSW model
The Lifshitz-Slyozov-Wagner (LSW) theory of domain coarsening describes the time evolution of the concentration \(c(\lambda,\tau)\geq 0\) at time \(\tau\geq 0\) of clusters of size \(\lambda\in (0,\infty)\) and reads \[ \begin{align*}{ \partial_\tau c(\lambda,\tau) & = - \partial_\lambda\left( \lambda^\beta \left( u(\tau)\ \lambda^{1/\nu} - 1 \right)\ c(\lambda,\tau) \right)\,, \cr u(\tau) & = \frac{\int_0^\infty \lambda^\beta\ c(\lambda,\tau)\ d\lambda}{\int_0^\infty \lambda^{\beta+1/\nu}\ c(\lambda,\tau)\ d\lambda}\,, }\end{align*} \] the latter condition guaranteeing the conservation through time evolution of the first moment of \(c\). The parameters satisfy \(\beta\in [0,1)\) and \(\nu\geq 1\), the classical LSW model corresponding to \(\beta=0\) and \(\nu=3\). It is by now well-known that the LSW equation has a one-parameter family of self-similar solutions, and their stability is of great importance to check the predictions of the theory. Contrary to what was originally suggested by Lifshitz \& Slyozov (1961), each self-similar solution has a rather large domain of attraction: the convergence of \(c(\tau)\) as \(\tau\to\infty\) towards a particular self-similar solution actually depends on the behaviour of the initial condition \(\lambda\mapsto c(\lambda,0)\) at the edge of its support if it is compactly supported or for large \(\lambda\). In the particular case \(\beta=0\) and \(\nu=1\), more precise information can be obtained thanks to a convenient representation formula for \(c\), and a complete characterisation of the domains of attraction of all self-similar solutions is given here, thereby extending [\textit{J.~Carr} and \textit{O.~Penrose}, Physica D 124, No.~1--3, 166--176 (1998; Zbl 0935.80003)]. As in the pioneering work of [\textit{B.~Niethammer} and \textit{R. L.~Pego}, J. Stat. Phys. 95, No.~5--6, 867--902 (1999; Zbl 1005.82510)], the characterisation involves slowly varying functions.
scientific article; zbMATH DE number 5073059
scientific article